blob: comparison blob1.tex

equal deleted inserted replaced

-:4ef2f77a4652
+:15e6335ff1d4
-\documentclass[11pt,leqno]{article}
+\documentclass[11pt,leqno]{amsart}
-\usepackage{amsmath,amssymb,amsthm}
+\newcommand{\pathtotrunk}{./}
+\input{text/article_preamble.tex}
-\usepackage[all]{xy}
+\input{text/top_matter.tex}
 % test edit #3
 %%%%% excerpts from my include file of standard macros
-\def\bc{{\cal B}}
+\def\bc{{\mathcal B}}
 \def\z{\mathbb{Z}}
 \def\r{\mathbb{R}}
 \def\c{\mathbb{C}}
 \def\t{\mathbb{T}}
 \newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}}
 % tricky way to iterate macros over a list
 \def\semicolon{;}
 \def\applytolist#1{
-	\expandafter\def\csname multi#1\endcsname##1{
+\expandafter\def\csname multi#1\endcsname##1{
-		\def\multiack{##1}\ifx\multiack\semicolon
+\def\multiack{##1}\ifx\multiack\semicolon
-			\def\next{\relax}
+\def\next{\relax}
-		\else
+\else
-			\csname #1\endcsname{##1}
+\csname #1\endcsname{##1}
-			\def\next{\csname multi#1\endcsname}
+\def\next{\csname multi#1\endcsname}
-		\fi
+\fi
-		\next}
+\next}
-	\csname multi#1\endcsname}
+\csname multi#1\endcsname}
 % \def\cA{{\cal A}} for A..Z
-\def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}}
+\def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}}
 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;
 % \DeclareMathOperator{\pr}{pr} etc.
 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign};
+\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign};
 %%%%%% end excerpt
 \makeatletter
 \@addtoreset{equation}{section}
 \gdef\theequation{\thesection.\arabic{equation}}
 \makeatother
-\newtheorem{thm}[equation]{Theorem}
-\newtheorem{prop}[equation]{Proposition}
-\newtheorem{lemma}[equation]{Lemma}
-\newtheorem{cor}[equation]{Corollary}
-\newtheorem{defn}[equation]{Definition}
 \maketitle
 \section{Introduction}
 (motivation, summary/outline, etc.)
 (motivation:
 (1) restore exactness in pictures-mod-relations;
 (1') add relations-amongst-relations etc. to pictures-mod-relations;
 (2) want answer independent of handle decomp (i.e. don't
 just go from coend to derived coend (e.g. Hochschild homology));
 (3) ...
 )
 \section{Definitions}
 \subsection{Fields}
 Fix a top dimension $n$.
 A {\it system of fields}
 \nn{maybe should look for better name; but this is the name I use elsewhere}
 is a collection of functors $\cC$ from manifolds of dimension $n$ or less
 to sets.
 These functors must satisfy various properties (see KW TQFT notes for details).
 For example:
 there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$;
 there is a restriction map $\cC(X) \to \cC(\bd X)$;
 gluing manifolds corresponds to fibered products of fields;
 given a field $c \in \cC(Y)$ there is a ``product field"
 $c\times I \in \cC(Y\times I)$; ...
 \nn{should eventually include full details of definition of fields.}
 \nn{note: probably will suppress from notation the distinction
 between fields and their (orientation-reversal) duals}
 \nn{remark that if top dimensional fields are not already linear
 then we will soon linearize them(?)}
 The definition of a system of fields is intended to generalize
 the relevant properties of the following two examples of fields.
 The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$
 is a manifold of dimension $n$ or less) to be the set of
 all maps from $X$ to $B$.
 The second example will take longer to explain.
 Given an $n$-category $C$ with the right sort of duality
 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
 we can construct a system of fields as follows.
 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
 with codimension $i$ cells labeled by $i$-morphisms of $C$.
 We'll spell this out for $n=1,2$ and then describe the general case.
 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
 an object (0-morphism) of the 1-category $C$.
 A field on a 1-manifold $S$ consists of
 \begin{itemize}
-	\item A cell decomposition of $S$ (equivalently, a finite collection
+\item A cell decomposition of $S$ (equivalently, a finite collection
 of points in the interior of $S$);
-	\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
+\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
 by an object (0-morphism) of $C$;
-	\item a transverse orientation of each 0-cell, thought of as a choice of
+\item a transverse orientation of each 0-cell, thought of as a choice of
 ``domain" and ``range" for the two adjacent 1-cells; and
-	\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
+\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
 domain and range determined by the transverse orientation and the labelings of the 1-cells.
 \end{itemize}
 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
 interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
 of the algebra.
 \medskip
 A field on a 0-manifold $P$ is a labeling of each point of $P$ with
 an object of the 2-category $C$.
 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
 A field on a 2-manifold $Y$ consists of
 \begin{itemize}
-	\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
+\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
 that each component of the complement is homeomorphic to a disk);
-	\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
+\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
 by a 0-morphism of $C$;
-	\item a transverse orientation of each 1-cell, thought of as a choice of
+\item a transverse orientation of each 1-cell, thought of as a choice of
 ``domain" and ``range" for the two adjacent 2-cells;
-	\item a labeling of each 1-cell by a 1-morphism of $C$, with
+\item a labeling of each 1-cell by a 1-morphism of $C$, with
 domain and range determined by the transverse orientation of the 1-cell
 and the labelings of the 2-cells;
-	\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
+\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
 to $\pm 1 \in S^1$; and
-	\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
+\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
 determined by the labelings of the 1-cells and the parameterizations of the previous
 bullet.
 \end{itemize}
 \nn{need to say this better; don't try to fit everything into the bulleted list}
 For general $n$, a field on a $k$-manifold $X^k$ consists of
 \begin{itemize}
-	\item A cell decomposition of $X$;
+\item A cell decomposition of $X$;
-	\item an explicit general position homeomorphism from the link of each $j$-cell
+\item an explicit general position homeomorphism from the link of each $j$-cell
 to the boundary of the standard $(k-j)$-dimensional bihedron; and
-	\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
+\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
 domain and range determined by the labelings of the link of $j$-cell.
 \end{itemize}
 %\nn{next definition might need some work; I think linearity relations should
 %be treated differently (segregated) from other local relations, but I'm not sure
 %the next definition is the best way to do it}
 \medskip
 For top dimensional ($n$-dimensional) manifolds, we're actually interested
 in the linearized space of fields.
 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is
 the vector space of finite
 linear combinations of fields on $X$.
 If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$.
 Thus the restriction (to boundary) maps are well defined because we never
 take linear combinations of fields with differing boundary conditions.
 In some cases we don't linearize the default way; instead we take the
 spaces $\cC_l(X; a)$ to be part of the data for the system of fields.
 In particular, for fields based on linear $n$-category pictures we linearize as follows.
 Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
 obvious relations on 0-cell labels.
 More specifically, let $L$ be a cell decomposition of $X$
 and let $p$ be a 0-cell of $L$.
 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
 Then the subspace $K$ is generated by things of the form
 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
 to infer the meaning of $\alpha_{\lambda c + d}$.
 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
 will do something similar below; in general, whenever a label lives in a linear
 space we do something like this; ? say something about tensor
 product of all the linear label spaces?  Yes:}
 For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
 Define an ``almost-field" to be a field without labels on the 0-cells.
 (Recall that 0-cells are labeled by $n$-morphisms.)
 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
 space determined by the labeling of the link of the 0-cell.
 (If the 0-cell were labeled, the label would live in this space.)
 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
 We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the
 above tensor products.
 \subsection{Local relations}
 Let $B^n$ denote the standard $n$-ball.
 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$
 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.
 \nn{Roughly, these are (1) the local relations imply (extended) isotopy;
 (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and
 (3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$).
 See KW TQFT notes for details.  Need to transfer details to here.}
 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
 where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
 Let $X$ be an $n$-manifold.
 Assume a fixed system of fields.
 In this section we will usually suppress boundary conditions on $X$ from the notation
 (e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$).
 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
 submanifold of $X$, then $X \setmin Y$ implicitly means the closure
 $\overline{X \setmin Y}$.
 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case.
 (See xxxx above.)
 \nn{maybe restate this in terms of direct sums of tensor products.}
 There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear
 combination of fields on $X$ obtained by gluing $r$ to $u$.
 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
 just erasing the blob from the picture
 (but keeping the blob label $u$).
 Note that the skein space $A(X)$
 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.
 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
 2-blob diagrams (defined below), modulo the usual linear label relations.
 \nn{and also modulo blob reordering relations?}
 \nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams}
 the unlabeled blob or 0-cell.
 Let $c = \lambda a + b$.
 Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly.
 Then we impose the relation
 \eq{
-	x_c = \lambda x_a + x_b .
+x_c = \lambda x_a + x_b .
 }
 \nn{should do this in terms of direct sums of tensor products}
 Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$
 of their blob labelings.
 Then we impose the relation
 \eq{
-	x = \sign(\pi) x' .
+x = \sign(\pi) x' .
 }
 (Alert readers will have noticed that for $k=2$ our definition
 of $\bc_k(X)$ is slightly different from the previous definition
 of $\bc_2(X)$ --- we did not impose the reordering relations.
 if removing $B_j$ creates new twig blobs.
 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$,
 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
 Finally, define
 \eq{
-	\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
+\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
 }
 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
 Thus we have a chain complex.
 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
 \nn{TO DO:
 expand definition to handle DGA and $A_\infty$ versions of $n$-categories;
 relations to Chas-Sullivan string stuff}
 \section{Basic properties of the blob complex}
 \begin{prop} \label{disjunion}
 There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
 \end{prop}
 \begin{proof}
 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
 blob diagram $(b_1, b_2)$ on $X \du Y$.
 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way.
 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines
 a pair of blob diagrams on $X$ and $Y$.
 \end{proof}
 For the next proposition we will temporarily restore $n$-manifold boundary
 conditions to the notation.
 Suppose that for all $c \in \cC(\bd B^n)$
 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
 of the quotient map
 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
 \nn{always the case if we're working over $\c$}.
 Then
 \begin{prop} \label{bcontract}
 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to
 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.
 \nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$}
 \end{proof}
 (Note that for the above proof to work, we need the linear label relations
 for blob labels.
 Also we need to blob reordering relations (?).)
 (Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.)
 %maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}
 \begin{prop}
 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
 of $n$-manifolds and diffeomorphisms to the category of chain complexes and
 (chain map) isomorphisms.
 \qed
 \end{prop}
 \nn{need to same something about boundaries and boundary conditions above.
 $X$ to get blob diagrams on $X\sgl$:
 \begin{prop}
 There is a natural chain map
 \eq{
-	\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
+\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
 }
 The sum is over all fields $a$ on $Y$ compatible at their
 ($n{-}2$-dimensional) boundaries with $c$.
 `Natural' means natural with respect to the actions of diffeomorphisms.
 \qed
 \end{prop}
 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
 and $X\sgl = X_1 \cup_Y X_2$.
 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
 For $x_i \in \bc_*(X_i)$, we introduce the notation
 \eq{
-	x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
+x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
 }
 Note that we have resumed our habit of omitting boundary labels from the notation.
 \bigskip
 \section{$n=1$ and Hochschild homology}
 In this section we analyze the blob complex in dimension $n=1$
 and find that for $S^1$ the homology of the blob complex is the
 Hochschild homology of the category (algebroid) that we started with.
 \nn{or maybe say here that the complexes are quasi-isomorphic?  in general,
 should perhaps put more emphasis on the complexes and less on the homology.}
 Notation: $HB_i(X) = H_i(\bc_*(X))$.
 Let us first note that there is no loss of generality in assuming that our system of
 fields comes from a category.
 (Or maybe (???) there {\it is} a loss of generality.
 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be
 thought of as the morphisms of a 1-category $C$.
 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$
 are $A(I; a, b)$, and composition is given by gluing.
 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change
 and neither does $A(I; a, b) = HB_0(I; a, b)$.
 Let $C$ be a *-1-category.
 Then specializing the definitions from above to the case $n=1$ we have:
 \begin{itemize}
 \item $\cC(pt) = \ob(C)$ .
 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.
 Then an element of $\cC(R; c)$ is a collection of (transversely oriented)
 points in the interior
 of $R$, each labeled by a morphism of $C$.
 The intervals between the points are labeled by objects of $C$, consistent with
 the boundary condition $c$ and the domains and ranges of the point labels.
 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by
 composing the morphism labels of the points.
 Note that we also need the * of *-1-category here in order to make all the morphisms point
 the same way.
 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single
 point (at some standard location) labeled by $x$.
 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the
 form $y - \chi(e(y))$.
 Thus we can, if we choose, restrict the blob twig labels to things of this form.
 \end{itemize}
 We want to show that $HB_*(S^1)$ is naturally isomorphic to the
 Hochschild homology of $C$.
 \nn{Or better that the complexes are homotopic
 or quasi-isomorphic.}
 In order to prove this we will need to extend the blob complex to allow points to also
 be labeled by elements of $C$-$C$-bimodules.
 \end{itemize}
 First we show that $F_*(C\otimes C)$ is
 quasi-isomorphic to the 0-step complex $C$.
 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
 the point $*$ is $1 \otimes 1 \in C\otimes C$.
 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.
 Fix a small $\ep > 0$.
 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex
 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from
 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$.
 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
 (See Figure xxxx.)
 Let $x \in F^\ep_*$ be a blob diagram.
 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to
 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
 Let $y_i$ be the restriction of $z_i$ to $B_\ep$.
 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
 Define $j_\ep(x) = \sum x_i$.
 \nn{need to check signs coming from blob complex differential}
 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also.
 The key property of $j_\ep$ is
 \eq{
-	\bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
+\bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
 }
 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field
 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$.
 Note that $\sigma_\ep(x) \in F'_*$.
 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$
 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.
 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.
 Instead, we'll be less ambitious and just show that
 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have
 $x \in F_*^\ep$.
 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of
 finitely many blob diagrams.)
 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map
 $F'_* \sub F_*(C\otimes C)$ is surjective on homology.
 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$
 and
 \eq{
-	\bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
+\bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
 }
 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
 \medskip
 $G''_* \to G'_*$ induces an isomorphism on $H_0$.
 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that
 for all $x \in F'_*$ we have
 \eq{
-	x - \bd h(x) - h(\bd x) \in F''_* .
+x - \bd h(x) - h(\bd x) \in F''_* .
 }
 Since $F'_0 = F''_0$, we can take $h_0 = 0$.
 Let $x \in F'_1$, with single blob $B \sub S^1$.
 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$.
 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$).
 \nn{need to say above more clearly and settle on notation/terminology}
 Finally, we show that $F''_*$ is contractible.
 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}
 Let $x$ be a cycle in $F''_*$.
 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
 ball $B \subset S^1$ containing the union of the supports and not containing $*$.
 Adding $B$ as a blob to $x$ gives a contraction.
 \nn{need to say something else in degree zero}
 This completes the proof that $F_*(C\otimes C)$ is
 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows.
 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if
 * is a labeled point in $y$.
 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *.
 Let $x \in \bc_*(S^1)$.
 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
 $x$ with $y$.
 It is easy to check that $s$ is a chain map and $s \circ i = \id$.
 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
 in a neighborhood $B_\ep$ of *, except perhaps *.
 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$.
 \nn{rest of argument goes similarly to above}
 \bigskip
 Does the above exactness and contractibility guarantee such a map without writing it
 down explicitly?
 Probably it's worth writing down an explicit map even if we don't need to.}
+We can also describe explicitly a map from the standard Hochschild
+complex to the blob complex on the circle. \nn{What properties does this
+map have?}
+\begin{figure}%
+$$\mathfig{0.6}{barycentric/barycentric}$$
+\caption{The Hochschild chain $a \tensor b \tensor c$ is sent to
+the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.}
+\label{fig:Hochschild-example}%
+\end{figure}
+As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly.
+The edges marked $x, y$ and $z$ carry the $1$-chains
+\begin{align*}
+x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\
+y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\
+z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab}
+\end{align*}
+and the $2$-chain labelled $A$ is
+\begin{equation*}
+A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}.
+\end{equation*}
+Note that we then have
+\begin{equation*}
+\bdy A = x+y+z.
+\end{equation*}
+In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations,
+$$\phi\left(\Tensor_{i=1}^n a_i) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$
+with ...
 \section{Action of $C_*(\Diff(X))$}  \label{diffsect}
 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of
 \nn{be more restrictive here?  does more need to be said?}
 \begin{prop}  \label{CDprop}
 For each $n$-manifold $X$ there is a chain map
 \eq{
-	e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) .
+e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) .
 }
 On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$
 (Proposition (\ref{diff0prop})).
 For any splitting $X = X_1 \cup X_2$, the following diagram commutes
 \eq{ \xymatrix{
-	 CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X}    & \bc_*(X) \\
+CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X}    & \bc_*(X) \\
-	 CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
-	 	\ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}}  \ar[u]^{\gl \otimes \gl}  &
+\ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}}  \ar[u]^{\gl \otimes \gl}  &
-			\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl}
+\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl}
 } }
 Any other map satisfying the above two properties is homotopic to $e_X$.
 \end{prop}
 The proof will occupy the remainder of this section.
 Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is
 {\it adapted to $\cU$} if there is a factorization
 \eq{
-	P = P_1 \times \cdots \times P_m
+P = P_1 \times \cdots \times P_m
 }
 (for some $m \le k$)
 and families of diffeomorphisms
 \eq{
-	f_i :  P_i \times X \to X
+f_i :  P_i \times X \to X
 }
 such that
 \begin{itemize}
 \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$;
 \item the $V_i$'s are mutually disjoint;
 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
 where $k_i = \dim(P_i)$; and
 \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$
 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.
 \end{itemize}
 A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum
 The proof will be given in Section \ref{fam_diff_sect}.
 \medskip
 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$
 (e.g.~the support of a blob diagram).
 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if
 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$
 either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$.
 A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells,
 each of which is compatible.
 (Note that we could strengthen the definition of compatibility to incorporate
 a factorization condition, similar to the definition of ``adapted to" above.
 The weaker definition given here will suffice for our needs below.)
 \begin{cor}  \label{extension_lemma_2}
 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$.
 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$.
 \end{cor}
 \begin{proof}
 This will follow from Lemma \ref{extension_lemma} for
 appropriate choice of cover $\cU = \{U_\alpha\}$.
 Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let
 $V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$.
 Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$
 either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$.
 Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$,
 with the (compatible) support of $f_i$ in place of $X$.
 This insures that the resulting homotopy $h_i$ is compatible.
 Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$.
 \nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this}
 \end{proof}
 later draft}
 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
 about boundary later}
 Recall that we are given
 an open cover $\cU = \{U_\alpha\}$ and an
 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
 Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
 As a first approximation to the argument we will eventually make, let's replace $x$
 with a single singular cell
 \eq{
-	f: P \times X \to X .
+f: P \times X \to X .
 }
 Also, we'll ignore for now issues around $\bd P$.
 Our homotopy will have the form
 \eqar{
-	F: I \times P \times X &\to& X \\
+F: I \times P \times X &\to& X \\
-	(t, p, x) &\mapsto& f(u(t, p, x), x)
+(t, p, x) &\mapsto& f(u(t, p, x), x)
 }
 for some function
 \eq{
-	u : I \times P \times X \to P .
+u : I \times P \times X \to P .
 }
 First we describe $u$, then we argue that it does what we want it to do.
 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$.
 The various $K_\alpha$ should be in general position with respect to each other.
 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
 which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
 For $p \in D$ we define
 \eq{
-	u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
+u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
 }
 (Recall that $P$ is a single linear cell, so the weighted average of points of $P$
 makes sense.)
 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
 For handles of $J$ of index less than $k$, we will define $u$ to
 interpolate between the values on $k$-handles defined above.
 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
 of $E$.
 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
 with a $k$-handle.
 Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell
 corresponding to $E$.
 Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$
 adjacent to the $k{-}1$-cell corresponding to $E$.
 For $p \in E$, define
 \eq{
-	u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
+u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
-			+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
++ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
 }
 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
 the $k{-}j$-cell corresponding to $E$.
 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
 Now define, for $p \in E$,
 \eq{
-	u(t, p, x) = (1-t)p + t \left(
+u(t, p, x) = (1-t)p + t \left(
-			\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
+\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
-				+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
++ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
-			 \right) .
+\right) .
 }
 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
 mentioned above.
 This completes the definition of $u: I \times P \times X \to P$.
 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions,
 then $F$ is a homotopy through diffeomorphisms.
 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
 We have
 \eq{
-%	\pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
+%   \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
-	\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
+\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
 }
 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
 \nn{bounded away from zero, or something like that}.
 (Recall that $X$ and $P$ are compact.)
 Also, $\pd{f}{p}$ is bounded.
 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$
 is a singular cell adapted to $\cU$.
 This will complete the proof of the lemma.
 \nn{except for boundary issues and the `$P$ is a cell' assumption}
 Let $j$ be the codimension of $D$.
 (Or rather, the codimension of its corresponding cell.  From now on we will not make a distinction
 between handle and corresponding cell.)
 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
 where the $j_i$'s are the codimensions of the $K_\alpha$
 cells of codimension greater than 0 which intersect to form $D$.
 It follows that the singular cell $D \to \Diff(X)$ can be written as a product
 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.
 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
 Next case: $j=2$, $m=1$, $j_1 = 2$.
 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
 1-dimensional.
 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
 and a 2-cell with support $U_\beta$.
 Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$.
 \section{$A_\infty$ action on the boundary}
 Let $Y$ be an $n{-}1$-manifold.
 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary
 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure
 of an $A_\infty$ category.
 Composition of morphisms (multiplication) depends of a choice of homeomorphism
 $I\cup I \cong I$.  Given this choice, gluing gives a map
 \eq{
-	\bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c)
+\bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c)
-			\cong \bc_*(Y\times I; a, c)
+\cong \bc_*(Y\times I; a, c)
 }
 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various
 higher associators of the $A_\infty$ structure, more or less canonically.
 \nn{is this obvious?  does more need to be said?}
 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$.
 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism
 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$
 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the
 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$.
 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood
 of $Y$ in $X$.
 In the next section we use the above $A_\infty$ actions to state and prove
 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy
 of $Y \du -Y$ contained in its boundary.
 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$.
 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex
 of $X$.
 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$,
 where $c\sgl \in \cC(\bd X\sgl)$,
 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation
 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$.
 \begin{thm}
 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product
 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$.
 \end{thm}
 The proof will occupy the remainder of this section.
 \nn{...}
 \section{Extension to ...}
 \nn{Need to let the input $n$-category $C$ be a graded thing
 (e.g.~DGA or $A_\infty$ $n$-category).}
 \nn{maybe this should be done earlier in the exposition?
 if we can plausibly claim that the various proofs work almost
 the same with the extended def, then maybe it's better to extend late (here)}
 %Recall that for $n$-category picture fields there is an evaluation map
 %$m: \bc_0(B^n; c, c') \to \mor(c, c')$.
 %If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain
 %map $m: \bc_*(B^n; c, c') \to \mor(c, c')$.

changeset 8	15e6335ff1d4
parent 7	4ef2f77a4652
child 10	fa1a8622e792